The present invention relates to an optimized design method of a holographic optical element so that an error caused by aberration of the holographic optical element can be reduced, and an apparatus using such a holographic optical element.
Holographic optical elements (HOE's) are small, light-weight, thin-films which have the potential for being very inexpensive in a mass production. The elements have multiple functions such as laser beam focusing and scanning, beam-splitting, and spectral selectivity. HOE application have been demonstrated in various fields. HOE's can be used to perform wave phase transformations. In designing HOE's, it is important to minimize outgoing wavefront aberrations.
This is because the hologram's reference wave conditions are often different from those of the reconstructed incident wave. Since sensitivity of a sensitive plate becomes highly responsive to a wavelength shorter than that of a semiconductor laser beam, the reference wave having a wavelength shorter than that of the semiconductor laser beam is used. For example, in HOE's incidence angle and wave length differ between the constructed and reconstructed waves. In order to minimize the aberrations, it is necessary to use optimization procedures for designing a holographic element having a complicated grating function. There has been a great deal of research concerning optimized design methods for hologram phase transfer functions. Fairchild used numerical iterative ray-tracing techniques (R. C. Fairchild and J. R. Fienup, "Computer-originated aspheric holographic optical elements", Opt. Eng. 21, 133-140 (1982)).
A holographic laser scanner was designed by applying a similar method (Ishii and K. Murata, "Flat-field linearized scans with reflection dichromated gelatin holographic gratings", APPl.OPT.23, 1999-2006 (1984)). For these designs, the optimized routine is performed with the Damped Least Squares (DLS) method, which is also used for lens design. However, the DLS method has the following disadvantages:
(1) Extensive calculations of ray directions are required. PA1 (2) Solutions often converge to local minima rather than to the desired absolute minimum. PA1 (3) It is difficult to determine whether an optimal solution has been attained. PA1 (1) The method must use the mean-squared wavefront aberration as the error function. PA1 (2) The method must be applicable to optimized design of laser scanners. PA1 (reconstructed light phase .PHI..sub.out)=(incident wave phase .PHI..sub.in)+(phase transfer function .PHI..sub.H); PA1 (wavefront aberration)=(reconstructed light phase)-(nonaberration reconstructed light phase). PA1 (phase transfer function)=(spheric term .PHI..sub.H(0))+(aspheric term).
Winick proposed analytical optimization of phase transfer functions (K. Winick and J. Fienup, "Optimium holographic elements recorded with aspheric wavefronts", J. Opt. Soc. Am. 73, 207-217 (1983)). This method uses the mean-squared wavefront aberration as the error function. However, a complicated procedure is required to design phase transfer functions of holograms and it is necessary to resort to approximate solutions. Several methods were also proposed which uses the Fourier transform hologram. It seems, however, that complicated calculation is required for these methods. As a result, such optimization procedures do not yield an exact solution except in specific cases. These methods were never applied to designing holographic scanners. For holographic scanners, there is a method that avoids the DLS routine. For this method a spot diagram is used for an evaluation which is not made on the basis of diffraction theory. However, optical system aberration is small, geometric optical evaluation is insufficient; diffraction theory evaluation is required. There have been no method which satisfies both of the following condictions:
The most difficult point is that the error function is a nonlinear equation of aspheric phase coefficients.